= θ θ
1
) ) ( ) (
(
z g z dz
p
tj ti tj , θ∈ ( −∞ , ∞ )
. (20) The parameter θ controls the complementarity or substitutability of individual innate ability. When θ = 1, Equation (20) is the same as Equation (2); thus peer quality is determined by the average innate ability of students within the same school.If θ >1, then the amount of school quality enhanced by students with high innate ability is more than the amount brought down by students with low ability. This implies that individual innate ability is more substitutable and this is referred to here as the ‘role model’ type of peer quality. On the other hand, if θ <1, then the amount of school quality brought down by students with low innate ability will be more than the amount raised by students with high ability. This is referred to as the ‘bad apple’
type of peer quality because it implies that individual innate ability is more complementary.15
4.1 ‘Role Model’ and ‘Bad Apple’ Peer Quality
We go on to explore the sensitivity of non-linear peer quality, analyzing both the ‘role model’ and ‘bad apple’ types of peer quality. The peer group effect (γ2) is assigned to its value in the baseline model (0.19). To study the impacts of the ‘role model’ type of peer quality, we set θ = 1.1, 1.8 and 2.5. To analyze the impacts of the ‘bad apple’
type of peer quality, we assign θ = 0.9, 0.2 and –0.5. For each type of peer quality, we study the case both with and without voucher programs with a scale equal to 50 per cent. Tables 2 and 3 present details of economic performance under non-linear peer quality at the end of the 4th period. The variables
p
andlp
represent non-linear peer quality and the mean ability (θ= 1
) within the public school in the 4th period, respectively.4.1.1 ‘Role model’ type of peer quality
Table 2 exhibits the simulation results with ‘role model’ type peer quality. The higher the magnitude of θ , the greater the substitutability for an individual’s ability; the
15 Following Snipes (1998), we use the term ‘role model’ and ‘bad apple’ type peer quality to represent the situation for θ > 1 and for θ < 1.
substitutability of each individual’s innate ability becomes clear when we compare
p
withlp
. As Table 2 shows,p
is higher thanlp
because high-ability students contribute more to school quality than low-ability students.Table 2 ‘Role model’ type of peer quality (
γ2 =0.19)
θ 1.1 1.8 2.5
= 0
υ υ
= 0 . 5 ∆ %
υ= 0
υ= 0 . 5 ∆ %
υ= 0
υ= 0 . 5 ∆ %
Public (%) 87.870 75.360 -14.237 88.460 76.710 -13.283 89.030 77.740 -12.681
p
2.241 1.997 -10.888 2.368 2.096 -11.486 2.491 2.188 -12.164lp
2.225 1.986 -10.742 2.238 2.008 -10.277 2.250 2.026 -9.956g
2.286 2.333 2.056 2.299 2.340 1.783 2.311 2.351 1.731 Gini (%) 35.259 36.607 3.823 35.004 36.413 4.025 34.766 36.194 4.107Note: * Definition of variables:
lp
= average innate ability of student body in the public school in the 4th period; for other variables: see Table 1.For each assigned value of θ , the implementation of voucher programs will reduce the enrollment rate and the peer quality in public schools because more able students and children with richer parents will switch to private schools. Consequently, there will be an increase in both the growth rate and income inequality.
In order to explore the sensitivity of non-linear peer group qualities, we examine the statistics under a mixed educational system, with no voucher program in place, when the substitutability of each individual’s innate ability is equal to 1.1, 1.8 and 2.5.
As θ increases, there is a rise in the magnitude of the contribution from high-ability students to public school quality, and thus, more families are willing to send their children to public schools. We find that as θ increases from 1.1, to 1.8, and finally to 2.5, the public school enrollment rate rises from 87.87 per cent, to 88.46 per cent, and then to 89.03 per cent. An increase in θ raises peer quality within a school and hence, also raises school quality. Therefore, an increase in θ will also lead to an increase in economic growth. With more students attending public schools due to an increase in θ , there is a corresponding decline in income inequality. A similar scenario can also be found when voucher programs are introduced.16
We now turn to an examination of the impacts of voucher programs for various levels of θ . The parameter
∆ %
represents the percentage change in the16 With vouchers at a scale of 50 per cent, as θ increases from 1.1, to 1.8, and 2.5, the respective public school enrollment rate increases from 75.36 per cent, to 76.71 per cent, and 77.74 per cent.
corresponding statistics, with and without vouchers. Comparing the economic growth before and after the implementation of voucher programs, we find that the higher the level of θ , the smaller the percentage reduction in the public school enrollment rate and the greater the level of income inequality. This is because when θ is high, the human capital accumulation of students remaining in public schools will be enhanced by able students, and hence, more students will remain in public schools after the implementation of voucher programs. Although an increase in θ implies that students in public schools can accumulate more human capital, with a lower percentage increase in the private school enrollment rate, this will reduce the overall increase in economic growth both before and after the implementation of voucher programs.17
4.1.2 ‘Bad apple’ type of peer quality
In contrast to the ‘role model’ type of peer quality, when θ is less than 1, school quality is brought down by students with low innate ability, largely offsetting the enhancements to school quality provided by students with high innate ability. The statistics on school quality under the ‘bad apple’ type of peer quality are provided in Table 3.
Table 3 ‘Bad apple’ type of peer quality (
γ2 =0.19)
θ 0.9 0.2 -0.5
= 0
υ υ
= 0 . 5 ∆ %
υ= 0
υ= 0 . 5 ∆ %
υ= 0
υ= 0 . 5 ∆ %
Public (%) 87.570 75.100 -14.240 86.720 73.250 -15.533 85.760 71.150 -17.036
p
2.202 1.970 -10.536 2.068 1.862 -9.961 1.934 1.751 -9.462lp
2.219 1.981 -10.726 2.200 1.951 -11.318 2.179 1.918 -11.978g
2.281 2.333 2.324 2.266 2.326 2.648 2.253 2.320 2.974 Gini (%) 35.369 36.621 3.537 35.685 36.855 3.279 36.019 37.067 2.910Note: * For definition of variables: see Tables 1 and 2.
The lower the magnitude of θ , the greater the level of complementarity will be for an individual’s ability. Due to the impacts of the ‘bad apple’ type of peer quality on school quality,
p
is smaller thanlp
, as shown in the table. For a fixed value of θ , the enrollment rate in public schools declines when vouchers are introduced.
17 This is because only those students with high parental human capital, or high ability, or both, will choose private schools to accumulate their human capital more rapidly than in public schools.
The complementary of innate ability within public schools will push high-ability students to leave public schools, thereby further worsening peer quality within public schools. Those attending private schools will accumulate more human capital whilst those staying in public schools will accumulate less human capital. If the former dominates the latter, then implementation of voucher programs will raise the growth rate, and vice versa. The simulation results show that the implementation of voucher programs will lower peer quality within public schools, whilst increasing both the growth rate and income inequality in all three cases.
Comparing the statistics under a mixed educational system, for different magnitudes of θ , and with no voucher program in place, we find that enrollment in public schools declines with a decrease in θ ; as θ declines from 0.9, to 0.2, and finally to –0.5, the public school enrollment rate declines from 87.57 per cent, to 86.72 per cent, and then to 85.76 per cent. A decrease in θ reduces peer quality within a school and hence, lowers school quality; as θ decreases, there is a corresponding reduction in the economic growth rate. With fewer students attending public schools, as θ decreases, there is a corresponding increase in income inequality.
The story is also very similar under a mixed educational system with voucher programs.18
We now turn to an examination of the impacts of voucher programs for various levels of θ. Comparing the economic growth rate before and after the implementation of voucher programs, we find that the smaller the level of θ , the greater the percentage increase in the public school enrollment rate and the economic growth rate.
This is because when θ is low, the human capital of students remaining in public schools will be lowered by students of low ability. Hence, the implementation of voucher programs will cause more students to leave public schools for private schools and this will, in turn, raise the economic growth rate.
Comparing Table 2 with Table 3, we find that with the same scale of vouchers, the public school enrollment rate is higher under ‘role model’ peer quality than under
‘bad apple’ peer quality. This implies that the public school is more preferable when peer quality is more substitutable, as opposed to when it is more complementary.
Figure 4 presents the economic growth rate and income inequality in the 4th period for
18 With vouchers at a scale of 50 per cent, as θ decreases from 0.9, to 0.2, and finally to -0.5, the respective public school enrollment rate declines from 75.10 per cent, to 73.25 per cent, and then to 71.15 per cent.
values of θ considered in Tables 1, 2 and 3 when there is no vouchers and when the scale of vouchers equals 5%. It shows that the growth rate goes up with an increase in θ while Gini coefficient goes down with an increase in θ .
Figure 4 Economic performance under different values of θ
5. CONCLUSIONS
A simple dynamic model is constructed in this paper, under a mixed educational system, to explore the impacts of voucher programs on economic performance when there are peer group effects. The government provides all public schooling, as well as tuition vouchers for households attending private schools. With the coexistence of public and private schools, the public/private school enrollment rate is an important determinant of the pattern of growth and inequality. Therefore, a policy of intervention in the educational system would seem to be a necessary element of government policy when targeted growth rates or inequality levels are being pursued, and thus, the impacts of educational reform will be of crucial concern for a government. The two most common instruments used by governments to affect an individual’s decision on education are income tax and vouchers. Changing the tax rate will affect the return on human capital whilst the scale of voucher programs will
determine the public/private school enrollment rate. This paper focuses on the second issue with an examination of several voucher programs.
This study differs from much of the literature relating to vouchers since it analyzes the impact of vouchers on growth and inequality when there are peer group effects. In addition to the problem of segregation, our computation results indicate that an increase in the scale of vouchers will raise income inequality, and suggests further that this is not always growth enhancing. With strong peer group effects, increasing the scale of vouchers will in fact lower the economic growth rate.
In addition to linear peer quality, we also examine cases for non-linear peer quality and find that the impacts of voucher programs on economic performance are sensitive both to the magnitude of the peer group effects in the human capital accumulation function, and the way in which peer interactions affect school quality.
Our study indicates that in any future study, a more accurate estimation of the human capital accumulation function and peer quality formulation will be necessary.